Title: Bi-Lipschitz embedding in Banach spaces, Rademacher-type theorems, and
functions of bounded variation Abstract: A mapping between metric spaces is L-bi-Lipschitz if it stretches
distances by a factor of at most L, and compresses them by a factor no worse
than 1/L. A basic problem in geometric analysis is to determine when one
metric space can be bi-Lipschitz embedded in another, and if so, to
estimate the optimal bi-Lipschitz constant. In recent years this
question has generated great interest in computer science, since many
data sets can be represented as metric spaces, and associated algorithms
can be simplified, improved, or estimated, provided one knows that
the metric space space in question can be bi-Lipschitz embedded (with
controlled bi-Lipschitz constant) in a nice space, such as L^2 or L^1.
The lecture will discuss several new existence and non-existence results
for bi-Lipschitz embeddings in Banach spaces. One approach to non-existence
theorems is based on generalized differentiation theorems in the spirit of
Rademacher's theorem on the almost everywhere differentiability of Lipschitz
functions on R^n. We first show that earlier differentiation based results
of Pansu and Cheeger, which proved non-existence of embeddings into R^k,
generalize to many Banach space targets, such as L^p, for 1 < p < infinity.
We then focus on the case when the target is L^1, where differentiation theory
is known to fail, and the embedding questions are of particular interest in computer
science. When the domain is the Heisenberg group with its Carnot-Caratheodory
metric, we show that a modified form of differentiation still holds for Lipschitz
maps into L^1, by exploiting a new connection with functions of bounded variation,
and some very recent advances in geometric measure theory. This lead to a proof
of a conjecture of Assaf Naor.
This is joint work with Jeff Cheeger.